TITLE: How variance is defined?
QUESTION [7 upvotes]: The variance of a random variable $X$ is defined as $E[(x-\mu )^2]$. Why can't it be defined as $E[|x-\mu |]$. i.e., What is the basic idea behind this definition.
Thank you.

REPLY [4 votes]: $\newcommand{\var}{\operatorname{var}}$
This seems to be a recurring question.
Maybe the most important part of the answer is that with the conventional definition of variance, if $X_1,\ldots,X_n$ are independent random variables, then
$$
\var(X_1+\cdots+X_n) = \var(X_1)+\cdots+\var(X_n).
$$
This makes it possible to know what the variance is of the number of heads that result from tossing a coin $20000$ times (or of any other sum of i.i.d. random variables), and thus to use the central limit theorem to find things like the probability that the number of heads is between $10,000\pm60$.